It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for the Apollo program leading to its incorporation in the Apollo navigation computer. This digital filter is sometimes called the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, non-linear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich.This Kalman filter was first described and partially developed in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). In fact, some of the special case linear filter's equations appeared in these papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow.The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a weighted average.The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more.Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems.The underlying model is a Bayesian model similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.
The Kalman filter deals effectively with the uncertainty due to noisy sensor data and to some extent also with random external factors.The weights are calculated from the covariance, a measure of the estimated uncertainty of the prediction of the system's state.The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone.A common application is for guidance, navigation, and control of vehicles, particularly aircraft and spacecraft.Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics.Due to the time delay between issuing motor commands and receiving sensory feedback, usage of the Kalman filter supports the realistic model for making estimates of the current state of the motor system and issuing updated commands. In the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties.